View source for Book:Thomas Ernst/A Comprehensive Treatment of q-Calculus

{{Book|A Comprehensive Treatment of q-Calculus|2012|Springer Basel||Thomas Ernst}}

===Contents===
:Introduction
::1.1 A survey of the chapters
::1.2 What is $q$-calculus?
:::1.2.1 Elementary series manipulation
::1.3 Update on $q$-calculus
:::1.3.1 Current textbooks on this subject
:::1.3.2 Comparison with complex analysis
::1.4 Comparison with nonstandard analysis
::1.5 Comparison with the units of physics
::1.6 Analogies between $q$-analysis and analysis
::1.7 The first $q$-functions
:2 The different languages of $q$-calculus
::2.1 Schools --- traditions
::2.2 Ramifications and minor Schools
:::2.2.1 Different notations
::2.3 Finite differences and Bernoulli numbers
::2.4 Umbral calculus, interpolation theory
::2.5 Elliptic and Theta Schools and notations, the oldest roots -- the $q$-forerunners
::2.6 Trigonometry, prosthaphaeresis, logarithms
::2.7 The development of calculus
::2.8 The Faulhaber mathematics
::2.9 Descartes, Leibniz, Hindenburg, Arbogast
::2.10 The Fakultäten
::2.11 Königsberg School
::2.12 Viennese School
::2.13 Göttingen School
::2.14 The combinatorial School: Gudermann, Grünert
::2.15 Heidelberg School
::2.16 Weierstraß, formal power series and the $\Gamma$ function
::2.17 Halle $q$-analysis School
::2.18 Jakob Friedrich Fries, Martin Ohm, Babbage, Peacock and Herschel
::2.19 Different styles in $q$-analysis
:3 Pre $q$-Analysis
::3.1 The early connection between analytic number theory and $q$-series
::3.2 Some aspects of combinatorial identities
:::3.2.1 Faà di Bruno formula
::3.3 The duality between Bernoulli and Stirling numbers
::3.4 Tangent numbers, Euler numbers
::3.5 The occurrence of binomial coefficient identities in the literature
::3.6 Nineteenth century: Catalna, Grigoriew, Imchenetsky
::3.7 A short history of hypergeometric series
:::3.7.1 The $\Gamma$ function
:::3.7.2 Balanced and well-poised hypergeometric series
:::3.7.3 Fractional differentiation
:::3.7.4 Newton, Taylor, Stirling, Montmort
:::3.7.5 Euler's contribution
:::3.7.6 Vandermonde and Pfaffian summation formulas
:::3.7.7 Conic sections in the seventeenth century
:::3.7.8 The infinity in England
:::3.7.9 The infinity in the hands of Euler
:::3.7.10 The infinity, the binomial coefficients
:::3.7.11 Gauß' contribution
:::3.7.12 After Gauß: Clausen, Jacobi
:::3.7.13 Kummer's contribution
:::3.7.14 Cauchy, Riemann, Heine, Thomae, Papperitz
:::3.7.15 1800-1914; Sonine, Goursat, Stieltjes, Schafheitlin, Pochhammer, Mellin
:::3.7.16 First half of the twentieth century; England, USA
:::3.7.17 Special functions defined by integrals
:::3.7.18 Second half of the twentieth century
::3.8 The Jacobi theta functions; different notations; properties
::3.9 Meromorphic continuation and Riemann surfaces
::3.10 Wave equation
::3.11 Orthogonal polynomials
:::3.11.1 Legendre-d'Allonville-Murphy polynomials
:::3.11.2 Laguerre-Abel-Sonine-Murphy-Chebyshev-Halphen-Szegő polynomials
:::3.11.3 Jacobi polynomials
:::3.11.4 Hermite polynomials
:4 The $q$-umbral calculus and semigroups. The Nørlund calculus of finite differences
::4.1 The $q$-umbral calculus and semigroups
::4.2 Finite differences
::4.3 $q$-Appell polynomials
:::4.3.1 The generalized $q$-Bernoulli polynomials
:::4.3.2 The Ward $q$-Bernoulli numbers
:::4.3.3 The generalized JHC $q$-Bernoulli polynomials
:::4.3.4 NWA $q$-Euler polynomials
:::4.3.5 The NWA generalized $q$-Euler numbers
:::4.3.6 Several variables; $n$ negative
:::4.3.7 $q$-Euler-Maclaurin expansions
:::4.3.8 JHC polynomials of many variables; negative order
:::4.3.9 JHC $q$-Euler-Maclaurin expansions
:::4.3.10 Euler symbolic formula
:::4.3.11 Complementary argument theorems
::4.4 $q$-Lucas and $q$-G polynomials
:::4.4.1 $q$-Lucas numbers
:::4.4.2 The $q$-G polynomials
:::4.4.3 Lucas and $G$ polynomials of negative order
:::4.4.4 Expansion formulas
::4.5 The semiring of Ward numbers
:5 $q$-Stirling numbers
::5.1 Introduction
::5.2 The Hahn-Cigler-Carlitz-Johnson approach
::5.3 The Carlitz-Gould approach
::5.4 The Jackson $q$-derivative as difference operator
::5.5 Applications
:6 The first $q$-functions
::6.1 $q$-analogue, $q$-factorial, tilde operator
:::[[Q-number|(6.1)]]
:::[[Q-number when a=n is a natural number|(6.2)]]
:::[[Q-factorial|(6.3)]]
:::(6.4)
:::(6.5)
:::(6.6)
:::(6.7)
:::(6.8)
:::(6.9)
:::(6.10)
:::(6.11)
:::(6.12)
:::(6.13)
:::(6.14)
:::(6.15)
:::(6.16)
:::(6.17)
:::(6.18)
:::(6.19)
:::(6.20)
::6.2 The $q$-derivative
::6.3 The $q$-integral
::6.4 Two other tilde operators
::6.5 The Gaussian $q$-binomial coefficients and the $q$-Leibniz theorem
:::6.5.1 Other formulas
::6.6 Cigler's operational method for $q$-identities
::6.7 Gould and Carlitz $q$-binomial coefficient identities
::6.8 $q$-Exponential and $q$-trigonometric functions
:::[[Q-exponential E sub q|6.150]]
:::[[Meromorphic continuation of q-exponential E sub q|6.151]]
:::[[Q-difference equation for q-exponential E sub q|6.152]]
:::[[Q-exponential E sub 1/q|6.153]]
:::[[Q-difference equation for q-exponential E sub 1/q|6.154]]
::6.9 The Heine function
::6.10 Oscillations in $q$-analysis
::6.11 The Jackson-Hahn-Cigler $q$-addition and $q$-analogues of the trigonometric functions
:::6.11.1 Further $q$-trigonometric functions
::6.12 The Nalli-Ward-Al-Salam $q$-addition and some variants of the $q$-difference operator
::6.13 Weierstraß elliptic functions and sigma functions
:::6.13.1 Elliptic functions
:::6.13.2 Connections with the $\Gamma_q$ function
::6.14 The Chen-Liu operator or parameter augmentation
:7 $q$-hypergeometric series
::7.1 Definition of the $q$-hypergeometric series
:::7.1.1 $q$-difference equation for ${}_{p+1}\phi_p$
::7.2 Balanced and well-poised $q$-hypergeometric series
::7.3 Advantages of the Heine definition
::7.4 $q$-Binomial theorem
::7.5 Jacobi's elliptic functions expressed as real and imaginary parts of $q$-hypergeometric series with exponential argument (Heine)
::7.6 The Jacobi triple product identity
::7.7 $q$-contiguity relations
::7.8 Heine $q$-transformations
:::7.8.1 The $q$-beta function
::7.9 Heine's $q$-analogue of the Gauß summation formula
::7.10 A $q$-analogue of the Pfaff-Saalschütz summation formula
::7.11 Sears' ${}_4 \phi_3$ transformation
::7.12 $q$-analogues of Thomae's transformations
::7.13 The Bailey-Daum summation formula
::7.14 A general expansion formula
::7.15 A summation formula for a terminating very-well poised ${}_4 \phi_3$ series
::7.16 A summation formula for a terminating very-well-poised ${}_6 \phi_5$ series
::7.17 Watson's transformation formula for a terminating very-well-poised ${}_8 \phi_7$ series
::7.18 Jackson's sum of a terminating very-well-poised balanced ${}_8 \phi_7$ series
:::7.18.1 Three corollaries
::7.19  Watson's proof of the Rogers-Ramanujan identities
::7.20 Bailey's 1929 transformation formula for a terminating, balanced, very-well-poised ${}_10 \phi_9$
::7.21 Watson's $q$-analogue of the Barnes contour integral
::7.22 Three $q$-analogues of the Euler integral formula for the function $\Gamma(x)$
::7.23 Inequalities for the $\Gamma_q$ function
::7.24 Summary of the umbral method
:8 Sundry topics
::8.1 Four $q$-summation formulas of Andrews
::8.2 Some quadratic $q$-hypergeometric transformations
::8.3 The Kummer ${}_2F_1(-1)$ formula and Jacobi's theta function
::8.4 Another proof of the $q$-Dixon formula
::8.5 A finite version of the $q$-Dixon formula
::8.6 The Jackson summation formula for a finite, $2$-balanced, well-poised ${}_5 \phi_4$ series
::8.7 The Jackson finite $q$-analogue of the Dixson formula
::8.8 Other examples of $q$-special functions
::8.9 $q$-analogues of two formulas by Brown and Eastham
::8.10 The $q$-analogue of Truesdell's function
::8.11 The Bailey transformation for $q$-series
::8.12 $q$-Taylor formulas with remainder; the mean value theorem 
:::8.12.1 The mean value theorem in $q$-analysis
::8.13 Bilateral series
::8.14 Fractional $q$-integrals
:9 $q$-orthogonal polynomials
::9.1 Ciglerian $q$-Laguerre polynomials
:::9.1.1 The different Laguerre-philosophies
:::9.1.2 The $q$-Laguerre polynomials
:::9.1.3 Generating functions and recurrences
:::9.1.4 Product expansions
:::9.1.5 Bilinear generating functions
:::9.1.6 Al-Salam operator expressions
:::9.1.7 The $q$-Laguerre Rodriguez operator
:::9.1.8 $q$-orthogonality
::9.2 $q$-Jacobi polynomials
:::9.2.1 Definition and the Rodriguez formula
:::9.2.2 The $q$-Jacobi Rodriguez operator
:::9.2.3 More generating functions and recurrences
:::9.2.4 $q$-orthogonality
::9.3 $q$-Legendre polynomials and Carlitz-Al-Salam polynomials
:::9.3.1 $q$-Legendre polynomials
:::9.3.2 Carlitz-Al-Salam polynomials
:10 $q$-functions of several variables
::10.1 The corresponding vector notation
::10.2 Historical introduction
::10.3 Transformations for basic double series
:::10.3.1 Double $q$-balanced series
:::10.3.2 Transformation formula of Carlitz-Srivastava
:::10.3.3 Three formulas of Andrews
:::10.3.4 $q$-Analogues of Carlson's formulas
::10.4 The $q$-Appell function $\Phi_1$ as $q$-integral
::10.5 $q$-analogues of some of Srivastava's formulas
::10.6 Some $q$-formulas of Srivastava
:::10.6.1 Generating functions
:::10.6.2 Transformations
:::10.6.3 Double sum identities (Srivastava and Jain)
::10.7 Two reduction formulas of Karlsson and Srivastava
::10.8 $q$-analogues of reducibility theorems of Karlsson
::10.9 $q$-Analogues of Burchnall-Chaundy expansions
:::10.9.1 $q$-analogues of Verma expansions
:::10.9.2 A similar formula
::10.10 Multiple extensions of the Rothe-von Grüson-Gauß formula
::10.11 An expansion formula in the spirit of Chaundy
::10.12 Formulas according to Burchnall-Chaundy and Jackson
:11 Linear partial $q$-difference equations
::11.1 Introduction
::11.2 Canoncial equations and symmetry techniques for $q$-series (Kalnins, Miller)
::11.3 $q$-difference equations for $q$-Appell and $q$-Lauricella functions
:12 $q$-Calculus and physics
::12.1 The $q$-Coulomb problem and the $q$-hydrogen atom
::12.2 Connections to knot theory
::12.3 General relativity
::12.4 Molecular and nuclear spectroscopy
::12.5 Elementary particle physics and chemical physics
::12.6 Electroweak interaction
::12.7 String theory
::12.8 Wess-Zumino model
::12.9 Quantum Chromodynamics
:References
:Index before 1900
:Index after 1900
:Name index before 1900
:Name index after 1900
:Name index Physics
:Notation index Chapter 1,2,6-9
:Notation index Chapter 3
:Notation index Chapter 4,5
:Notation index Chapter 10-11
:Notation index Chapter 12




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